Unitarity and causality constraints in composite Higgs models

We study the scattering of longitudinally polarized W bosons in extensions of the Standard Model where anomalous Higgs couplings to gauge sector and higher order O(p^4) operators are considered. These new couplings with respect to the Standard Model should be thought as the low energy remnants of some new dynamics involving the electroweak symmetry breaking sector. By imposing unitarity and causality constraints on the WW scattering amplitudes we find relevant restrictions on the possible values of the new couplings and the presence of new resonances above 300 GeV. We investigate the properties of these new resonances and their experimental detectability. Custodial symmetry is assumed to be exact throughout and the calculation avoids using the Equivalence Theorem as much as possible.


I. INTRODUCTION
Providing tools to assess the nature of the Higgs-like boson discovered at the LHC [1,2] is probably the most urgent task that theorist face in our time. New runs will in due time clarify whether the Higgs particle is truly elementary or there is a new scale of compositeness associated to it. In the latter case there should be a new strongly interacting sector, an extension of the Standard Model (SM) that conventionally is termed the extended electroweak symmetry breaking sector (EWSBS). All evidence suggests that the scale possibly associated to the EWSBS may be substantially larger than the electroweak scale v = 246 GeV, but it should not go beyond a few TeV. Otherwise the mass of its lightest scalar resonance becomes unnatural and very difficult to sustain [3].
Of course the Higgs could be elementary and the Minimal Standard Model (MSM) realized in nature but then some fundamental questions of elementary particle physics would remain unanswered: there would be no natural dark matter candidate -not even an axion, no hope of understanding the flavour puzzle and perhaps even the vacuum of the theory be unstable and jeopardize our whole picture of the universe (see [4] for updated results).
Effective Lagrangians of Higgs and gauge bosons have already extensively used to study current LHC data [5,6] combining also in some case LEP and flavour data. This approach has the advantage to be model-independent but the drawback is that number of operators is usually large and the choice of a convenient basis is subject of intense debate [7]. Here we are only interested in W W scattering and work in the custodial limit. Therefore, only a restrict number of operators have to be considered. The effective Lagrangian is Here the w are the three Goldstone of the global group SU(2) L × SU(2) R → SU(2) V . This symmetry breaking is the minimal pattern to provide the longitudinal components to the W ± and Z and emerging from phenomenology. Here, the Higgs field h is a gauge and SU(2) L × SU(2) R singlet. Larger symmetry group could be adopted [8] and consequently further Goldstone bosons may exist -the Higgs might be one of them. But all them eventually should acquire masses, drop from an extended unitary matrix U and could be parameterized by a polynomial expansion. The operators O i include the complete set of O(p 4 ) operators defined in [9,10]. Of these only two O 4 and O 5 will contribute to W L W L scattering in the custodial limit: where V µ = (D µ U) U † . When writing eq. (1) we have assumed the well-established chiral counting rules to limit the number of operators to the O(p 4 ) ones.
The parameters a and b control the coupling of the Higgs to the gauge sector [8]. Couplings containing higher powers of h/v do not enter W W scattering and they have not been included in (1). We have also introduced two additional parameters d 3 , and d 4 that parame- see [6,12] 2 . The present data clearly favours values of a close to the MSM value, while the a 4 and a 5 are still largely unbounded. The parameter b is almost totally undetermined at present. Other very important parameters are a 1 , a 2 and a 3 , entering the oblique and triple gauge coupling. Bounds on the oblique corrections are quite constraining [13], while the triple electroweak gauge coupling has already been measured with a level of precision [14] similar to LEP. Some results on the γγW + W − coupling are also available [15].
When a and b depart from their MSM values a = b = 1 the theory becomes unrenormalizable in the conventional sense, although at the one-loop level W L W L scattering can be rendered finite by a suitable redefinition of the coefficients a 4 and a 5 and a (together with v, H H and λ). The relevant counter-terms have been worked out in [11,16] using the Equivalence Theorem [17,18] (i.e. replacing longitudinally polarized W L and Z L by the 1 We bear in mind that this is not the most general form of the Higgs potential and in fact additional counterterms are needed beyond the Standard Model [11] but this does not affect the subsequent discussion for W L W L scattering 2 Our a and a 4,5 coefficients stand for a = 1 − ξc H /2, a 4 = ξ 2 c 11 and a 5 = ξc 6 of ref. [6]. c H range comes from the values of Set A in table 4 and c 6,11 are from table 8 of ref. [6]. corresponding Goldstone bosons w and z). This approximation is appropriate to obtain the relevant counter-terms for W L W L scattering and in [19] the renormalization is being extended to the remaining a i counter-terms (i = 4, 5).
In this work we extend the previous analysis [10] of unitarized W L W L scattering to the case a = 1 and b = 1, namely anomalous Higgs couplings to the gauge sector are now considered. More specifically, we will vary a, b as well as the a 4,5 parameters within the experimental bounds of eq. (4). We use the Inverse Amplitude Method (IAM) [20] to enforce the unitarity of longitudinally polarized W W amplitudes up to the O(p 4 ). The calculation of the amplitude is done avoiding the use of the Equivalence Theorem as much as possible.
The reason for this is that at the relatively low energies we are considering, the replacement of the W L and Z L by w and z is problematic in order to make accurate predictions. In the next sections, we will give examples of how misleading the ET can be if the right kinematical conditions are not met.
As in the previous work [10], we found that new resonances can appear in the parameter space of a 4,5 for given values of a and b even though for values of a > 1 the allowed region is drastically reduced by the causality constraint. More specifically, for a ≤ 1 and b free, the overall picture is very similar to one in [10] where the case a = b = 1 (experimentally favoured so far) was studied. In the scalar channel for example, new resonances go from masses as low as 300 GeV to nearly as high as the cutoff of the method of 4πv ≃ 3 TeV, with rather narrow widths typically from 10 to 100 GeV. In the vector channel the lowest achievable masses range from about 600 GeV up to the cutoff, with widths going from 5 to about 50 GeV. For a > 1, the picture is drastically different with respect the one in [10], since for a large portion of the a 4,5 parameter space many resonances have negative widths breaking causality.
It is usually expected that a new strongly interacting sector would lead to resonances in different channels but what turned out to be a bit of a surprise in our previous work [10] and in the present for a < 1 is that these resonances are typically narrow and very hard to detect. This appears to be directly related to the unitarization of the W L W L scattering in the presence of light Higgs. Searching for these resonances at LHC will be however very important because if none of them reveals itself below ∼ 3 TeV virtually all a 4,5 parameter space of the anomalous couplings could be excluded. This can be an indirect way of assessing these quartic electroweak boson couplings. Actually, no direct information on a 4 and a 5 exists at present from direct measurements of the quartic electroweak boson couplings.
Unfortunately the actual signal strength of the new resonances predicted is such that they are not currently being probed in LHC Higgs search data and consequently no relevant bounds on a 4 and a 5 can be derived at present from the existing data -a situation that may change soon. The previous considerations emphasize the importance of indirect measures of the couplings a 4 and a 5 by searching for the additional resonances coming out from our study of W LW L scattering. Measuring these anomalous couplings will be one of the main tasks of the LHC run starting in 2015.

II. ISOSPIN AND PARTIAL WAVE AMPLITUDES
Here we introduce the basic definition of our observables. We shall consistently assume our treatment that custodial symmetry is exactly preserved. This implies taking g ′ = 0 and ignoring all the O i operators that can contribute to W W scattering but O 4 and O 5 . This approximation also allows for a neat usage of the isospin formalism and for the convergence of the partial wave amplitudes. We also disregard operators that contain matter fields as they are totally irrelevant for the present discussion.
As emphasized in [10] when dealing with longitudinally polarized amplitudes, as opposed to using the ET approximation, caution must be exercised to account for an ambiguity introduced by the longitudinal polarization vectors that do not transform under Lorentz transformations as 4-vectors. Expressions involving the polarization vector ǫ µ L can not be cast in terms of the Mandlestam variables s, t, and u until an explicit reference frame has been chosen, as they can not themselves be written solely in terms of covariant quantities.
Obviously amplitudes still satisfy crossing symmetries when they remain expressed in terms of the external 4-momenta. A short discussion on this point is placed in appendix B.
, can be written using isospin and Bose symmetries as The fixed-isospin amplitudes are given by T 0 (s, t, u) = 00|S|00 = 3A +−00 + A ++++ (7) We shall also need the amplitude for the process W + W − → hh. Taking into account that the final state is an isospin singlet and defining the projection of this amplitude to the I = 0 channel gives The partial wave amplitudes for fixed isospin I and total angular momentum J are where the P J (x) are the Legendre polynomials and t = (1 − cos θ)(4M 2 − s)/2, u = (1 + cos θ)(4M 2 −s)/2 with M being the W mass. We will concern ourselves with only the lowest non-zero partial wave amplitude in each isospin channel: t 00 (s), t 11 (s), and t 20 (s), namely the scalar/isoscalar, vector/isovector, and isotensor amplitudes respectively. Unitarity directly implies that |t IJ (s)| < 1. For further implications of unitarity on t IJ (s) the interested reader may see ref. [21].
In this work, the partial wave amplitude t IJ (s) are studied up to O(p 4 ), namely Here t    IJ (s) by the one-loop expression of A +−00 calculated in ref. [16].   For values of a different from 1, the W L W L scattering amplitudes exhibit rather different behaviour with respect to the MSM case a = 1. The most important difference is that the |t IJ | < 1 unitarity bound is violated at tree-level pretty quickly. We shall see later how to restore unitarity with the help of higher loops and counter-terms but in this section we concentrate on the peculiarities of the tree level amplitudes t   IJ absent in ET approximation. Some of these features will be crucial in our analysis.
In order to study the behaviour of t (0) IJ , we will establish three different regions according to the range of the values of a = 1, a > 1 and a < 1.

A. Case a = 1
In Figure 2 we plot the tree-level isoscalar partial wave amplitude t (0) 00 (s) for W L W L → Z L Z L as a function of s. The external W legs are taken on-shell (p 2 = M 2 = M 2 W = M 2 Z ). As we see from Figure 2 the partial wave amplitude has a rather rich analytic structure. It has one pole at s = M 2 H but also a second singularity can be seen at the value s = 3M 2 . A closer examination reveals also a third singularity at s = 4M 2 − M 2 H , invisible in the Figure 2 as it happens to be multiplied by a very small number. These singularities correspond to poles of the t and u channel diagrams in Figure 1 that after the angular integration of eq. (10) to obtain the partial wave amplitudes behave as logarithmic divergences. The t and u channels are absent in the ET approximation. Note that both singularities are below the physical threshold at s = 4M 2 . Beyond the s = 3M 2 singularity the amplitude for a = 1 is always positive as can be seen in Figure 2.
In Figure 2 we also plot the tree-level partial wave amplitude t 11 amplitudes are also compared with the respective amplitudes obtained in ET approximation. As can be seen the ET is grossly inadequate at low energies. In particular it fails in reproducing the rich analytic structure of the amplitudes. The non-analyticity at s = 3M 2 and s = 4M 2 − M 2 H due to sub-threshold singularities is actually also present in the t (0) 20 partial wave amplitude (not depicted), corresponding like in the other two cases to a (zero width) logarithmic pole. In t should render the partial wave amplitude actually non-singular 3 .

B. Case a > 1
The three sub-threshold singularities appearing at a = 1 are also present in this case.
However, for a > 1 the partial wave amplitudes also show a new features. First of all, as shown in Figure 3 for a = 1.1 and amplitudes t   In addition, for a > 1 the tree-level partial wave amplitudes for t

IV. UNITARITY CORRECTIONS
In the case of Higgs anomalous couplings to gauge sector (a = 1 and b = 1) the tree-level amplitudes t (0) IJ are not-unitarity and we are forced to include additional operators in the theory, such as the a i counter-terms in eq. (1). At one-loop level, the a i will cancel the divergences of the Lagrangian in eq. (1) and finite couplings renormalized at some UV scale will remain [11,16], namely where f is the scale of the new interactions, and possibly other finite pieces.
Up to now, the calculation of the one-loop t IJ (s) contribution in eq. (11) is not available for a and b arbitrary and longitudinally polarized W and Z. This would require the evaluation of over one thousand diagrams. A numerical calculation is only available in [22] for the case a = b = 1 but it is not very useful for our purposes.
For this reason, to estimate the t IJ (s) contribution in eq. (11) we proceed in the following way. The analytic contribution from a 4,5 terms are calculated exactly with longitudinally polarized W and Z (appendix A) like the tree-level contribution t IJ (s) will however be determined using the ET [17,18]; i.e. we replace this loop amplitude by the corresponding process w + w − → zz. For this part of the calculation we take q 2 = 0 for external legs and set M = 0 but the Higgs mass is kept. The relevant diagrams of A(ww → zz) entering t (4) IJ (s) were calculated in [16] where explicit expressions for the different diagrams for arbitrary values of the couplings a and b can be found. This calculation has been checked and extended in [11], albeit setting M H = 0. As to the imaginary part of t (2) IJ (s) we can take advantage of the optical theorem to circumvent the problem of using the ET approximation. In the I = 1, J = 1 and I = 2, J = 0 cases we can use the relations Im t While for the I = 0 amplitude we also have a contribution from a two-Higgs intermediate

state. Then
Im t 00 (s) = σ(s)|t 00 (s)| 2 + σ H (s)|t H,0 (s)| 2 , with We believe that for the purpose of identifying dynamical resonances, normally occurring at s ≫ M 2 H the approximation of relying on the ET for the real part of the loops is fine. Note that the dominant contribution to the real part for large s, of order s 2 , is controlled by the contribution coming from couplings a 4,5 . We have also actually checked that, unless a 4 and a 5 are both very small, the contribution from the real part of the loop amounts only to a small correction to t (2) IJ . The final ingredient we need is a procedure to construct an unitary amplitude that perturbatively coincides with the tree plus one-loop result but incorporates the principle of unitarity. To this purpose, we use the Inverse Amplitude Method (IAM) [20] to the ampli-tude in eq.(11), namely which is identical to the [1,1] Padé approximant to t IJ derived from (11). The above expression obviously reproduces the first two orders of the perturbative expansion (eq. 11)and, in addition, satisfies the necessary unitarity constraints, namely |t IJ | < 1 at high energies and when the perturbative ingredients satisfy Im t as they must from the optical theorem. We refer to [10] and references therein for a more detailed discussion. We also recommend to read the recent article [23] for a rather complete review. In what follows we shall adhere to the procedure outlined in [10].
There is no really unambiguous way of applying the IAM to the case where there are coupled channels with different thresholds. This will be relevant to us only in the t 00 case as there is an intermediate state consisting in two Higgs particles. Here we shall adhere to the simplest choice that consists in assuming (17) to remain valid also in this case. In addition, there is decoupling of the two I = 0 channels in the case a 2 = b, as also discussed in [23] in the context of the ET approximation. We have checked our results for different values of b, in particular we see that setting b = a 2 does not give for the resonances that are eventually found results that are noticeably different from those obtained for other values of b. Finally, we have also checked explicitly the unitarity of our results.

V. LOOKING FOR RESONANCES IN a 4 AND a 5 PARAMETER SPACE
Non-renormalizable models such as the effective theory described by the Lagrangian (1) typically produce scattering amplitudes that grow too fast with the scattering energy breaking the unitarity bounds [21] at some point or other.
Chiral descriptions of QCD [24] are archetypal examples of this behavior and unitarization techniques have to be used to recover unitarity. The IAM [20], described in the previous section, is a convenient way of doing so. In QCD when the physical value of the pion decay constant f π and the O(p 4 ) low energy terms L i (as defined e.g. in [24], the counterpart of the a i in strong interactions) are inserted in the chiral Lagrangian and the IAM method is used, the validity of the chiral expansion is considerably extended and one is able to reproduce the ρ meson pole as well as many other properties of low energy QCD [20]. To find resonances, we perform a scan for the values |a 4 | < 0.01 and |a 5 | < 0.01 and a and b fixed looking for the presence or otherwise of resonances in the different channels. We will consider the different cases for a = 1 since the case a = 1 was discussed in detail in [10].
When looking for resonances we use two different methods. First we look for a zero of the real part of the denominator in (17)  We start by considering this case where the unitarized amplitudes t for the parameters a 4 , a 5 are therefore not acceptable. The presence of this excluded region is in exact correspondence with was found for the a = 1 case in [10] (and also with a similar situation in pion physics [20]).
In Fig. 6 we show the region of parameter space in a 4 , a 5 where isoscalar and isovector resonances exist for the value a = 0.9 along with the isotensor exclusion region. The pattern here has some analogies with the case a = 1 studied in [10] but proper 5 resonances are 4 Other values of a have also been studied but we here present results only for these two. 5 Recall that resonances are required to have, in addition to the correct causal properties, Γ < M/4. somewhat harder to form, in particular in the vector channel no resonance is found below 600 GeV for a = 0.9 in contrast to the a = 1 case studied in [10]. as these regions are, they are noticeably larger than the one corresponding to a = 1, which was virtually non-existent. This is true even for a = 0.95 which is very close to the MSM value for a, a = 1.
We have also considered the case where b = a 2 . In this case in the ET approximation the two channels decouple and our implementation of the IAM becomes better justified. Results for the case a = 0.9 and b = a 2 are depicted in Fig. 8. Changes with respect to b = 1 are unnoticeable indicating that b is of little relevance for the presence of resonances. Figure 9 shows the masses and widths of the scalar and vector resonances obtained for a = 0.9. As we see, in general they tend to be heavier and broader than the ones in the a = 1 case studied in [10]. We emphasize that the resonance in the scalar channel is additional to the Higgs at 125 GeV. The impact of parameter b is actually more visible in the widths of As we have seen the a < 1 case is really a smooth continuation of the a = 1 limit.
Resonances are somewhat more rare and they tend to be slightly heavier and broader, the more so as one departs from a = 1 but the modifications are small. This changes when we go to the a > 1 case. In this case the tree level amplitudes t for 1 < a < 1.125 they possess several additional zeroes, which disappear for a > 1.125. In the isovector channels the additional zeroes remain for even larger values of a. Past these zeroes, the tree-level contribution is negative all the way up to the limit of validity of the effective theory.
One finds zeroes of the denominator in eq. (17) that would correspond to resonances provided that the numerator does not vanish. This comment is relevant because many of the resonances present, particularly in the vector channel, appear in region near the last (as s increases) zero of the amplitude and this requires particular care. In fact for a set of values of a 4 and a 5 the determination as to whether a pole exists or not becomes ambiguous.
When we continue our amplitudes into their second Riemann sheet to estimate the width and solve for the complex pole we find that in various channels the imaginary part is such that it corresponds to a negative width. When two poles in a given channel are found, one is acceptable but then the other one leads to acausal behaviour (this can be proven analytically). For other values of the coupling the resonances are perfectly acceptable.
As an example of the pathologies found we show for a = 1.3 in Figure 12 the phase shifts for isotensor-scalar and isovector channel. We can see a behaviour that is incompatible with causality for the isotensor-scalar phase shift; recall that Γ = 2( dδ d √ s ) −1 . Sometimes a bona fide resonance pole coexists with a second resonance having negative width. This can be seen for instance in Fig. 11 in the scalar channel for a = 1.1. We see that one genuine looking resonance coexists with a huge singularity having a large negative width. The corresponding effective theory is unacceptable.
The net result is that a very sizeable part of the space of parameters is ruled out. For instance in Fig. 13 we show the excluded areas for a = 1.1, Fig. 13(a) and a = 1.3, Fig. 13(b).
We have seen that pathologies abound in the a > 1 case. In particular we have been unable to find a bona fide I = 2 resonance for a = 1.1 and a = 1.3 and this seems to be the generic situation for a > 1. This result is at odds with a recent dispersion relation analysis [26] claiming that theories having a > 1 must show a dominance of the I = 2 channel and even a model with a I = 2 resonance is suggested. An explanation for the discrepancy is given in appendix D.

VI. EXPERIMENTAL VISIBILITY OF THE RESONANCES
One thing is having a resonance and a very different one is being able to detect it. In particular the statistics so far available from the LHC experiments is limited. Searching for new particles in the LHC environment is extremely challenging. Yet a particle with the properties of the Higgs has been found with only limited statistics. This has been possible in part because of a fortunate upwards statistical fluctuation but also because the couplings and other properties of the Higgs were well known in the MSM. This is not necessarily the case for new resonances they may exist in the EWSBS. Fortunately the IAM method is able not only of predicting masses and widths but also their couplings to the W L W L and Z L Z L channels. In [10], where the case a = 1 was considered, the experimental signal of the different resonances was compared to that of a MSM Higgs with an identical mass. Because the decay modes are similar (in the vector boson channels that is) and limits on different Higgs masses are well studied this is a practical way of presenting the results.
Therefore in order to gain some intuition as to whether any of the predicted resonances for a < 1 should have been seen by now at the LHC we compare their signal (the size of the corresponding Breit-Wigner resonance) with the one of the Higgs at an equivalent mass. Just to gain some intuition on this we have used the easy-to-implement Effective W Approximation, or EWA [27]. The results are depicted in Fig. 14 for the W L W L and Z L Z L vector fusion channels. Note that both production modes are sub-dominant at the LHC with respect to gluon production mediated by a top-quark loop and also note that the decay modes of the resonances can be predicted with the technology presented here only for W L W L and Z L Z L final states.
What can be seen in these figures is that the signal is always lower than the one for a Higgs boson of an equivalent mass. However, the ratio σ resonance /σ Higgs seems to depend substantially on the value of a. For instance, for a = 1 it was found that in the scalar channel resonances, taken in the peak region as defined in [10]. Ratio of the Z L Z L scattering cross section due to dynamical resonances with that of the SM with a Higgs boson of the same mass for a scalar resonance (c). The LHC energy has been taken to be 8 TeV and the EWA approximation is assumed.
this ratio was typically lower than 0.1 and only in some very limited sector of parameter space could be as large as 0.3. It was even lower for ZZ production. Now for a = 0.9 we see that 0.2 is a more typical value for σ resonance /σ Higgs and in some areas of parameter space can go up to ∼ 0.4 or even close to 0.5. Again the signal is somewhat lower in the Z L Z L production channel. For the vector channel and again normalizing to the Higgs signal we get ratios for σ resonance /σ Higgs the signal ranges from 0.03 to 0.3.

VII. CONCLUSIONS
In this paper we have extended the analysis of [10] to the case a = 1 and b = 1 imposing the require of unitarity on the fixed isospin amplitudes contributing to longitudinal W scattering. The method chosen to unitarize the partial waves is the Inverse Amplitude Method. The simplicity of this method makes it suitable to analyze the problem being considered, while its validity has been well tested in strong interactions in the past.
We have seen that even in the presence of a light Higgs, it can help constrain anomalous couplings by helping predict heavier resonances, present in an extended EWSBS. The results for a = 1 presented here turn out to be partly in line with the results for a = 1 previously obtained if a < 1 and partly qualitatively different if a > 1 . If a < 1 for a large subset of values of the higher dimensional coefficients resonances are present. Typically they tend to be heavier and broader than in the a = 1 case. but only moderately so. They are never like the broad resonances that were entertained in the past in Higgsless models and this is undoubtedly a consequence of the unitarization that the presence of the Higgs brings about.
There is a smaller room for new states once unitarity is required. The properties of the resonance are therefore radically different from the initial expectations concerning W L W L scattering Current LHC Higgs search results do not yet probe the IAM resonances, but it may be possible in the near future, this is particularly true if a departs from its Standard Model value a = 1 because the resonances become higher and broader in this case with values for the ratio to the experimental signal that a Higgs with an equivalent mass would give σ resonance /σ Higgs can get close to 0.5 (recall that this applies only to the longitudinal vector gauge boson fusion channel). In any case it seems that LHC@14 TeV will be able to probe a reasonable part of the possible parameter space for resonances.
If resonances are found with the properties predicted here this discovery would immediately indicate that there is an extended EWSBS and that this is likely described by some strongly interacting theory, giving credit to the hypothesis of the Higgs being a composite state -most likely a pseudo-Goldstone boson. It would also provide immediate information on the value of some of the higher dimensional coefficients in the effective theory, probably much earlier that direct W L W L scattering would allow for a determination of the quartic gauge boson coupling.
We have also found another interesting result, namely that in the present framework theories with a > 1 are nearly excluded as the IAM predicts that they lead to resonances that violate causality in a large part of parameter space, the more so as one departs more from a = 1..

Appendix B: The issue of crossing symmetry
We would like to clarify the issue of crossing symmetry of amplitudes with external W L 's.
To this end let us consider just the tree-level contribution in the MSM to the processes W + L W − L → W + L W − L and W + L W + L → W + L W + L , respectively. To keep the formulae simple while making the point let us consider the limit s → ∞, −t → ∞ in the first process, which is consistent except for cos θ ≃ 1, and expand in M 2 /s and M 2 /t. We borrow the results from [18]. The resulting amplitude is The reason is that while crossing certainly holds when exchanging the external four vectors, the reference frame in which the two above amplitudes are expressed are different. In both cases they correspond to center-of-mass amplitudes, but after the exchange of momenta the two systems are boosted one with respect to the other. Writing the amplitudes in terms of s, t, u gives the false impresion that these expressions hold in any reference system but this is not correct because the polarization vectors are no true four-vectors.
On the contrary, the amplitudes computed via the ET are manifestly crossing symmetric because they amount to replacing ǫ µ L → k µ , which is obviously a covariant 4-vector. We insist once more that crossing does hold in any case but is not manifest for the scattering of longitudinal W bosons at the level of Mandelstam variables.

Appendix C: The origin of the logarithmic poles
Here we discuss the origins of the 3 singularities at s 0 = M 2 H , s 1 = 4M 2 − M 2 H and s 2 = 3M 2 ) entering the t IJ (s) amplitudes. These singularities can be tracked back from the terms 1/(s − M 2 H ), 1/(t − M 2 ) and 1/(u − M 2 ) in the W + L W − L → Z L Z L amplitude in eq. A4. The origin of the pole at s 0 is fairly obvious and needs no justification.
The origin of the pole at s 1 for t IJ (s) amplitudes is more complicated to see. First of all, let us notice that the fixed-isospin amplitudes T I in eq. 7 are combinations of the A ++00 = A(W + L W − L → Z L Z L ) in eq. A4 and its crossed amplitude A ++++ = A(W + L W + L → W + L W + L ). At this point, the term 1/(s − M 2 H ) in A ++00 , eq. A4, trasforms for the crossed amplitude A ++++ into 1/(t − M 2 H ) = (1/(−1 + cos θ)(−4M 2 + s)/2 − M 2 H ). Then, for cos θ = −1 we have a pole at s 1 and under integration on cos θ the amplitude t IJ (s) gets a logarithmic pole at s 1 .
Note that these singularities are all below threshold. Note too that except for s 0 they are absent in the ET treatment. For the LHC they appear at values of s corresponding to the replacement 4M 2 → i q 2 i as the external W are typically off-shell.